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The presence of task constraints imposes a significant challenge to motion planning. Despite all recent advancements, existing algorithms are still computationally expensive for most planning problems. In this paper, we present Constrained Motion Planning Networks (CoMPNet), the first neural planner for multimodal kinematic constraints. Our approach comprises the following components: i) constraint and environment perception encoders; ii) neural robot configuration generator that outputs configurations on/near the constraint manifold(s), and iii) a bidirectional planning algorithm that takes the generated configurations to create a feasible robot motion trajectory. We show that CoMPNet solves practical motion planning tasks involving both unconstrained and constrained problems. Furthermore, it generalizes to new unseen locations of the objects, i.e., not seen during training, in the given environments with high success rates. When compared to the state-of-the-art constrained motion planning algorithms, CoMPNet outperforms by order of magnitude improvement in computational speed with a significantly lower variance.